Predicting germination response to temperature. III. Model validation under field-variable temperature conditions

BACKGROUND AND AIMS: Two previous papers in this series evaluated model fit of eight thermal-germination models parameterized from constant-temperature germination data. The previous studies determined that model formulations with the fewest shape assumptions provided the best estimates of both germination rate and germination time. The purpose of this latest study was to evaluate the accuracy and efficiency of these same models in predicting germination time and relative seedlot performance under field-variable temperature scenarios. METHODS: The seeds of four rangeland grass species were germinated under 104 variable-temperature treatments simulating six planting dates at three field sites in south-western Idaho. Measured and estimated germination times for all subpopulations were compared for all models, species and temperature treatments. KEY RESULTS: All models showed similar, and relatively high, predictive accuracy for field-temperature simulations except for the iterative-probit-optimization (IPO) model, which exhibited systematic errors as a function of subpopulation. Highest efficiency was obtained with the statistical-gridding (SG) model, which could be directly parameterized by measured subpopulation rate data. Relative seedlot response predicted by thermal time coefficients was somewhat different from that estimated from mean field-variable temperature response as a function of subpopulation. CONCLUSIONS: All germination response models tested performed relatively well in estimating field-variable temperature response. IPO caused systematic errors in predictions of germination time, and may have degraded the physiological relevance of resultant cardinal-temperature parameters. Comparative indices based on expected field performance may be more ecologically relevant than indices derived from a broader range of potential thermal conditions.     

Predicting germination response to temperature. II. Three-dimensional regression, statistical gridding and iterative-probit optimization using measured and interpolated-subpopulation data

BACKGROUND AND AIMS: Most current thermal-germination models are parameterized with subpopulation-specific rate data, interpolated from cumulative-germination-response curves. The purpose of this study was to evaluate the relative accuracy of three-dimensional models for predicting cumulative germination response to temperature. Three-dimensional models are relatively more efficient to implement than two-dimensional models and can be parameterized directly with measured data. METHODS: Seeds of four rangeland grass species were germinated over the constant-temperature range of 3 to 38 degrees C and monitored for subpopulation variability in germination-rate response. Models for estimating subpopulation germination rate were generated as a function of temperature using three-dimensional regression, statistical gridding and iterative-probit optimization using both measured and interpolated-subpopulation data as model inputs. KEY RESULTS: Statistical gridding is more accurate than three-dimensional regression and iterative-probit optimization for modelling germination rate and germination time as a function of temperature and subpopulation. Optimization of the iterative-probit model lowers base-temperature estimates, relative to two-dimensional cardinal-temperature models, and results in an inability to resolve optimal-temperature coefficients as a function of subpopulation. Residual model error for the three-dimensional model was extremely high when parameterized with measured-subpopulation data. Use of measured data for model evaluation provided a more realistic estimate of predictive error than did evaluation of the larger set of interpolated-subpopulation data. CONCLUSIONS: Statistical-gridding techniques may provide a relatively efficient method for estimating germination response in situations where the primary objective is to estimate germination time. This methodology allows for direct use of germination data for model parameterization and automates the significant computational requirements of a two-dimensional piece-wise-linear model, previously shown to produce the most accurate estimates of germination time.     

Development of a Threshold Model to Predict Germination of Populus tomentosa Seeds after Harvest and Storage under Ambient Condition

Effects of temperature, storage time and their combination on germination of aspen (Populus tomentosa) seeds were investigated. Aspen seeds were germinated at 5 to 30°C at 5°C intervals after storage for a period of time under 28°C and 75% relative humidity. The effect of temperature on aspen seed germination could not be effectively described by the thermal time (TT) model, which underestimated the germination rate at 5°C and poorly predicted the time courses of germination at 10, 20, 25 and 30°C. A modified TT model (MTT) which assumed a two-phased linear relationship between germination rate and temperature was more accurate in predicting the germination rate and percentage and had a higher likelihood of being correct than the TT model. The maximum lifetime threshold (MLT) model accurately described the effect of storage time on seed germination across all the germination temperatures. An aging thermal time (ATT) model combining both the TT and MLT models was developed to describe the effect of both temperature and storage time on seed germination. When the ATT model was applied to germination data across all the temperatures and storage times, it produced a relatively poor fit. Adjusting the ATT model to separately fit germination data at low and high temperatures in the suboptimal range increased the models accuracy for predicting seed germination. Both the MLT and ATT models indicate that germination of aspen seeds have distinct physiological responses to temperature within a suboptimal range.     

Comparison of logistic regression and linear regression in modeling percentage data

Percentage is widely used to describe different results in food microbiology, e.g., probability of microbial growth, percent inactivated, and percent of positive samples. Four sets of percentage data, percent-growth-positive, germination extent, probability for one cell to grow, and maximum fraction of positive tubes, were obtained from our own experiments and the literature. These data were modeled using linear and logistic regression. Five methods were used to compare the goodness of fit of the two models: percentage of predictions closer to observations, range of the differences (predicted value minus observed value), deviation of the model, linear regression between the observed and predicted values, and bias and accuracy factors. Logistic regression was a better predictor of at least 78% of the observations in all four data sets. In all cases, the deviation of logistic models was much smaller. The linear correlation between observations and logistic predictions was always stronger. Validation (accomplished using part of one data set) also demonstrated that the logistic model was more accurate in predicting new data points. Bias and accuracy factors were found to be less informative when evaluating models developed for percentage data, since neither of these indices can compare predictions at zero. Model simplification for the logistic model was demonstrated with one data set. The simplified model was as powerful in making predictions as the full linear model, and it also gave clearer insight in determining the key experimental factors.     

Comparison of Logistic Regression and Linear Regression in Modeling Percentage Data

Percentage is widely used to describe different results in food microbiology, e.g., probability of microbial growth, percent inactivated, and percent of positive samples. Four sets of percentage data, percent-growth-positive, germination extent, probability for one cell to grow, and maximum fraction of positive tubes, were obtained from our own experiments and the literature. These data were modeled using linear and logistic regression. Five methods were used to compare the goodness of fit of the two models: percentage of predictions closer to observations, range of the differences (predicted value minus observed value), deviation of the model, linear regression between the observed and predicted values, and bias and accuracy factors. Logistic regression was a better predictor of at least 78% of the observations in all four data sets. In all cases, the deviation of logistic models was much smaller. The linear correlation between observations and logistic predictions was always stronger. Validation (accomplished using part of one data set) also demonstrated that the logistic model was more accurate in predicting new data points. Bias and accuracy factors were found to be less informative when evaluating models developed for percentage data, since neither of these indices can compare predictions at zero. Model simplification for the logistic model was demonstrated with one data set. The simplified model was as powerful in making predictions as the full linear model, and it also gave clearer insight in determining the key experimental factors.     

Predicting germination in semi-arid wildland seedbeds II. Field validation of wet thermal-time models

Accurate prediction of germination for species used for semi-arid land revegetation would support selection of plant materials for specific climatic conditions and sites. Wet thermal-time models predict germination time by summing progress toward germination subpopulation percentages as a function of temperature across intermittent wet periods or within singular wet periods. Wet periods may be defined by any reasonable seedbed water potential above which seeds are expected to imbibe sufficiently to germinate. These models may be especially applicable to the Artemisia steppe of the western U.S.A. where water availability limits germination in summer and early fall while cool temperatures limit germination in late fall, winter, and spring when soil water is available. To test accuracy of wet thermal-time models we placed seedbags with seeds of five species commonly used in wildland revegetation, as well as two collections of the invasive annual grass, Bromus tectorum L. into Artemisia tridentata Nutt. ssp. wyomingensis Beetle and Young zone seedbeds for 19 field incubation periods over four seasons. Hourly surface (1–3 cm) soil temperatures and soil water potentials were measured near the seedbags. These data were input into thermal-time models which predicted time to germination for each seedbag retrieval date. Binomial data representing agreement (1) or lack of agreement (0) of predicted and actual germination for each retrieval date were analyzed using logistic regression. Thermal summation method, season, water potential threshold, and species most affected accuracy of predictions (P < 0.0002). A model which defined a wet period as ≥−1.5 MPa soil water potential and summed progress toward germination across intermittent wet periods was most accurate in predicting actual germination by a retrieval date. Across all species, this model correctly predicted that germination would occur in seedbags 75–95% of the time over the latewinter to mid-spring seasons, but only 50–71% of the time for the fall-early winter season when time of soil water availability was least. Although the wet thermal-time model overestimated time to germination for some species and seasons, its accuracy should be high enough to evaluate germination potential by mid-spring for different species, sites, and climatic conditions.     

How well can the accuracy of comparative protein structure models be predicted?

Comparative structure models are available for two orders of magnitude more protein sequences than are experimentally determined structures. These models, however, suffer from two limitations that experimentally determined structures do not: They frequently contain significant errors, and their accuracy cannot be readily assessed. We have addressed the latter limitation by developing a protocol optimized specifically for predicting the Calpha root-mean-squared deviation (RMSD) and native overlap (NO3.5A) errors of a model in the absence of its native structure. In contrast to most traditional assessment scores that merely predict one model is more accurate than others, this approach quantifies the error in an absolute sense, thus helping to determine whether or not the model is suitable for intended applications. The assessment relies on a model-specific scoring function constructed by a support vector machine. This regression optimizes the weights of up to nine features, including various sequence similarity measures and statistical potentials, extracted from a tailored training set of models unique to the model being assessed: If possible, we use similarly sized models with the same fold; otherwise, we use similarly sized models with the same secondary structure composition. This protocol predicts the RMSD and NO3.5A errors for a diverse set of 580,317 comparative models of 6174 sequences with correlation coefficients (r) of 0.84 and 0.86, respectively, to the actual errors. This scoring function achieves the best correlation compared to 13 other tested assessment criteria that achieved correlations ranging from 0.35 to 0.71.     

Parameter estimation and goodness-of-fit in log binomial regression

An estimate of the risk, adjusted for confounders, can be obtained from a fitted logistic regression model, but it substantially over-estimates when the outcome is not rare. The log binomial model, binomial errors and log link, is increasingly being used for this purpose. However this model's performance, goodness of fit tests and case-wise diagnostics have not been studied. Extensive simulations are used to compare the performance of the log binomial, a logistic regression based method proposed by Schouten et al. (1993) and a Poisson regression approach proposed by Zou (2004) and Carter, Lipsitz, and Tilley (2005). Log binomial regression resulted in "failure" rates (non-convergence, out-of-bounds predicted probabilities) as high as 59%. Estimates by the method of Schouten et al. (1993) produced fitted log binomial probabilities greater than unity in up to 19% of samples to which a log binomial model had been successfully fit and in up to 78% of samples when the log binomial model fit failed. Similar percentages were observed for the Poisson regression approach. Coefficient and standard error estimates from the three models were similar. Rejection rates for goodness of fit tests for log binomial fit were around 5%. Power of goodness of fit tests was modest when an incorrect logistic regression model was fit. Examples demonstrate the use of the methods. Uncritical use of the log binomial regression model is not recommended.     

Predicting germination response of four cool-season range grasses to field-variable temperature regimes

Germination of non-dormant seeds under variable-temperature conditions can be predicted from constant-temperature germination data if it is assumed that instantaneous germination rate is independent of thermal history. Thermal-response models of this type have not been validated under simulated field-variable temperature conditions that vary in diurnal pattern, diurnal range and longer-term trends in mean–daily temperature. The purpose of this experiment was to evaluate germination response of thickspike wheatgrass (Elymus lanceolatus), bluebunch wheatgrass (Pseudoroegneria spicata), Sandberg bluegrass (Poa sandbergii) and bottlebrush squirreltail (Elymus elymoides) under both constant and field-variable temperature regimes in the laboratory. It was hypothesized that the thermal history assumption was valid and that constant-temperature data could be used to accurately estimate field-variable temperature response. Seeds were germinated at seven constant temperatures between 5 and 35°C, and under 18 variable-temperature regimes simulating six planting dates at three field sites. Predictions of germination time under variable-temperature conditions were accurate to within a fraction of 1 day up to a cumulative germination percentage of 70% for thickspike wheatgrass, 60% for bluebunch wheatgrass, 55% for Sandberg bluegrass and 70% for bottlebrush squirreltail. It was concluded that, for the variable-temperature regimes tested in this experiment, the thermal-history assumption was valid for earlier-germinating subpopulations.     

Isolating the cow-specific part of residual energy intake in lactating dairy cows using random regressions

The ability to properly assess and accurately phenotype true differences in feed efficiency among dairy cows is key to the development of breeding programs for improving feed efficiency. The variability among individuals in feed efficiency is commonly characterised by the residual intake approach. Residual feed intake is represented by the residuals of a linear regression of intake on the corresponding quantities of the biological functions that consume (or release) energy. However, the residuals include both, model fitting and measurement errors as well as any variability in cow efficiency. The objective of this study was to isolate the individual animal variability in feed efficiency from the residual component. Two separate models were fitted, in one the standard residual energy intake (REI) was calculated as the residual of a multiple linear regression of lactation average net energy intake (NEI) on lactation average milk energy output, average metabolic BW, as well as lactation loss and gain of body condition score. In the other, a linear mixed model was used to simultaneously fit fixed linear regressions and random cow levels on the biological traits and intercept using fortnight repeated measures for the variables. This method split the predicted NEI in two parts: one quantifying the population mean intercept and coefficients, and one quantifying cow-specific deviations in the intercept and coefficients. The cow-specific part of predicted NEI was assumed to isolate true differences in feed efficiency among cows. NEI and associated energy expenditure phenotypes were available for the first 17 fortnights of lactation from 119 Holstein cows; all fed a constant energy-rich diet. Mixed models fitting cow-specific intercept and coefficients to different combinations of the aforementioned energy expenditure traits, calculated on a fortnightly basis, were compared. The variance of REI estimated with the lactation average model represented only 8% of the variance of measured NEI. Among all compared mixed models, the variance of the cow-specific part of predicted NEI represented between 53% and 59% of the variance of REI estimated from the lactation average model or between 4% and 5% of the variance of measured NEI. The remaining 41% to 47% of the variance of REI estimated with the lactation average model may therefore reflect model fitting errors or measurement errors. In conclusion, the use of a mixed model framework with cow-specific random regressions seems to be a promising method to isolate the cow-specific component of REI in dairy cows.     

Validation of a calibrated prediction model for response to growth hormone treatment in an independent cohort

BACKGROUND: Prediction models, e.g. for prediction of response to growth hormone treatment, need validation in appropriate independent cohorts, comparing predicted and observed outcomes. In a previous validation of a model for predicting the first-year response to growth hormone treatment in children with idiopathic growth hormone deficiency, overfitting was observed. We modified the prediction formula and now report validation of this modified model. PATIENTS AND METHODS: The modified and original prediction models were applied to a group of patients selected from Lilly's GeNeSIS database using the same inclusion and exclusion criteria as for the original model. For both prediction methods, observed first-year height velocity was plotted vs. predicted height velocity in a calibration plot. For a valid prediction, the regression line should correspond to the line of identity (observed outcome is equal to predicted outcome); the regression lines for each prediction model were tested for significant differences from this line of identity. RESULTS: The number of patients fulfilling the criteria was 226. The regression line in the calibration plot of the modified model was not significantly different from the line of identity (p = 0.43), in contrast to the original model (p < 0.001). For the modified model the mean (SD) prediction error was -0.11 (2.05) cm/year and for the original model 0.28 (2.11) cm/year. CONCLUSION: The modified prediction method, obtained after calibration of the original model, performs well in an independent patient sample and gives more accurate predictions than the original model.     

Two statistical approaches to forecasting the start and duration of the pollen season of Ambrosia in the area of Lyon (France)

The aim of the present study was to forecast the start and duration of the pollen season of Ambrosia from meteorological data, in order to provide early information to allergists and allergic people. We used the airborne pollen data from Lyon (France), sampled using a Hirst trap from 1987 to 1999, and the meteorological data for the same period: air temperature (minimal, maximal, and average), rainfall, relative humidity, sunshine duration and soil temperature. Two forecasting models were used, one summing the temperatures and the other making use of a multiple regression on 10-day or monthly meteorological parameters. The start of the pollen season was predicted with both methods, results being more accurate with the regression (the errors between the predicted and the observed SDP ranging from 0 to 3 days). The duration of the pollen season was predicted by a regression model, errors ranging from 0 to 7 days. The models were later tested with satisfactory results from 2 additional years (2000 and 2001). Such forecasting models are helpful for allergic people, who have to begin their anti-allergic treatment before the start of the pollen season and not when the symptoms have appeared, since a preventive treatment is more efficient than a curative one. The regression allows predictions to be made 3-5 weeks in advance and so it is of particular interest. The forecasts will be broadcast on the Internet.     

Robustness of Generalized Estimating Equation (GEE) Tests of Significance against Misspecification of the Error Structure Model

Generalized linear model analyses of repeated measurements typically rely on simplifying mathematical models of the error covariance structure for testing the significance of differences in patterns of change across time. The robustness of the tests of significance depends, not only on the degree of agreement between the specified mathematical model and the actual population data structure, but also on the precision and robustness of the computational criteria for fitting the specified covariance structure to the data. Generalized estimating equation (GEE) solutions utilizing the robust empirical sandwich estimator for modeling of the error structure were compared with general linear mixed model (GLMM) solutions that utilized the commonly employed restricted maximum likelihood (REML) procedure. Under the conditions considered, the GEE and GLMM procedures were identical in assuming that the data are normally distributed and that the variance‐covariance structure of the data is the one specified by the user. The question addressed in this article concerns relative sensitivity of tests of significance for treatment effects to varying degrees of misspecification of the error covariance structure model when fitted by the alternative procedures. Simulated data that were subjected to monte carlo evaluation of actual Type I error and power of tests of the equal slopes hypothesis conformed to assumptions of ordinary linear model ANOVA for repeated measures except for autoregressive covariance structures and missing data due to dropouts. The actual within‐groups correlation structures of the simulated repeated measurements ranged from AR(1) to compound symmetry in graded steps, whereas the GEE and GLMM formulations restricted the respective error structure models to be either AR(1), compound symmetry (CS), or unstructured (UN). The GEE‐based tests utilizing empirical sandwich estimator criteria were documented to be relatively insensitive to misspecification of the covariance structure models, whereas GLMM tests which relied on restricted maximum likelihood (REML) were highly sensitive to relatively modest misspecification of the error correlation structure even though normality, variance homogeneity, and linearity were not an issue in the simulated data.Goodness‐of‐fit statistics were of little utility in identifying cases in which relatively minor misspecification of the GLMM error structure model resulted in inadequate alpha protection for tests of the equal slopes hypothesis. Both GEE and GLMM formulations that relied on unstructured (UN) error model specification produced nonconservative results regardless of the actual correlation structure of the repeated measurements. A random coefficients model produced robust tests with competitive power across all conditions examined. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of the error between experimental tetanic force curves of MUs of rat medial gastrocnemius muscle and their models by summation of equal successive contractions

More accurate muscle models require appropriate modelling of individual twitches of motor units (MUs) and their unfused tetanic contractions. It was shown in our previous papers, using a few MUs, that modelling of unfused tetanic force curves by summation of equal twitches is not accurate, especially for slow MUs. The aim of this study was to evaluate this inaccuracy using a statistical number of MUs of the rat medial gastrocnemius muscle (15 of slow, 15 of fast resistant and 15 of fast fatigable type). Tetanic contractions were evoked by trains of 41 stimuli at random interpulse intervals and different mean frequencies, resembling discharge patterns observed during natural muscle activity. The tetanic curves were calculated by the summation of equal twitches according to the respective experimental patterns. The previously described 6-parameter analytical function for twitch modelling was used. Comparisons between the experimental and the modelled curves were made using two coefficients: the fit coefficient and the area coefficient. The errors between modelled and experimental tetanic forces were substantially different between the three MU types. The error was the most significant for slow MUs, which develop much higher forces in real contractions than could be predicted based on the summation of equal twitches, while the smallest error was observed for FF MUs – their recorded tetanic forces were similar to those predicted by modelling. The obtained results indicate the importance of the inclusion of the type-specific non-linearity in the summation of successive twitch-like contractions of MUs in order to increase the reliability of modelling skeletal muscle force.     

A review of morphology-elasticity relationships in human trabecular bone: theories and experiments

In the perspective of predicting mechanical from morphological properties of human trabecular bone, the theoretical and experimental relationships between volume fraction, fabric and elastic properties were reviewed.Five data sets of human trabecular bone and two data sets of idealized cells were obtained from various investigators and analyzed statistically with one isotropic and four anisotropic models. For each model, multiple linear regressions were performed to fit the components of both the compliance and the stiffness tensors using volume fraction and in some cases fabric. The adjusted coefficients of determination of the regressions and the average relative errors of the reported versus the predicted tensor norms were calculated. The three anisotropic models that implied a log transformation of the data showed the best results. Excluding the idealized cell data, the adjusted coefficients of determination of these models ranged from 0.80 to 0.95 for the compliance and from 0.80 to 0.94 for the stiffness tensors, while the average relative errors varied between 16% and 55% for the compliance and between 25% and 62% for the stiffness data. The use of volume fraction alone in the isotropic model decreased the adjusted coefficients of determination by 0.03-0.25 and increased the average relative errors by 5-27%.This review confirms the potential of morphology-elasticity relationships for estimation of elastic properties of human trabecular bone using peripheral quantitative computed tomography or magnetic resonance imaging, but emphasizes the need for standardized measurements of mechanical properties at both continuum and tissue level.     

Dynamic variability in thermal-germination response of squirreltail (Elymus elymoides and Elymus multisetus)

Bottlebrush squirreltail (Elymus elymoides) and big squirreltail (Elymus multisetus) have been identified as high-priority species for restoration and rehabilitation of millions of acres of rangeland in the western United States that have been degraded by wildfire and introduced annual weeds. In this study, squirreltail accessions from Idaho, Colorado, Utah, Arizona and New Mexico were grown in a nursery environment to produce seeds in two different years for germination evaluation at 11 constant temperatures. A statistical-gridding model was used to predict cumulative germination rate of each seedlot for eight simulated planting dates between 1 January and 28 May over a 38-year seedbed-microclimatic simulation. Predicted germination response under simulated conditions of field-variable temperatures yielded a broader ecological basis for the relative ranking of thermal response than was obtained from single-value germination indices derived from either constant-temperature experiments, or from analysis of thermal-time coefficients.     

Optimum Sorption Isotherm by Linear and Nonlinear Methods for Safranin onto Alkali-Treated Rice Husk

Rice husk, a lignocellulosic by-product of the agroindustry, was treated with alkali and used as a low-cost adsorbent for the removal of safranin from aqueous solution in batch adsorption procedure. In order to estimate the equilibrium parameters, the equilibrium adsorption data were analyzed using the following two-parameter isotherms: Freundlich, Langmuir, and Temkin. A comparison of linear and nonlinear regression methods in selecting the optimum adsorption isotherm was applied on the experimental data. Six linearized isotherm models (including four linearized Langmuir models) and three nonlinear isotherm models are thus discussed in this paper. In order to determine the best-fit isotherm predicted by each method, seven error functions namely, coefficient of determination (r ²), the sum of the squares of the errors (SSE), sum of the absolute errors (SAE), average relative error (ARE), hybrid fractional error-function (HYBRID), Marquardt's percent standard deviation (MPSD), and the chi-square test (χ²) were used. It was concluded that the nonlinear method is a better way to obtain the isotherm parameters and the data were in good agreement with the Langmuir isotherm model.     

Self-consistent estimation of inter-residue protein contact energies based on an equilibrium mixture approximation of residues.

Pairwise contact energies for 20 types of residues are estimated self-consistently from the actual observed frequencies of contacts with regression coefficients that are obtained by comparing "input" and predicted values with the Bethe approximation for the equilibrium mixtures of residues interacting. This is premised on the fact that correlations between the "input" and the predicted values are sufficiently high although the regression coefficients themselves can depend to some extent on protein structures as well as interaction strengths. Residue coordination numbers are optimized to obtain the best correlation between "input" and predicted values for the partition energies. The contact energies self-consistently estimated this way indicate that the partition energies predicted with the Bethe approximation should be reduced by a factor of about 0.3 and the intrinsic pairwise energies by a factor of about 0.6. The observed distribution of contacts can be approximated with a small relative error of only about 0.08 as an equilibrium mixture of residues, if many proteins were employed to collect more than 20,000 contacts. Including repulsive packing interactions and secondary structure interactions further reduces the relative errors. These new contact energies are demonstrated by threading to have improved their ability to discriminate native structures from other non-native folds.